Multi-Allelic Gillespie-Sato Diffusion Models and their Extension to Infinite Allelic Ones

We first consider a multi-allelic Markov chain model for which one-step transition consists of three stages — independent reproduction, mutation and random sampling. Taking account of difference among alleles in means and variances of offspring numbers we discuss a diffusion approximation of the Markov chain model both in a finite-allelic case and in a countably infinite-allelic case. This diffusion approximation was derived by Gillespie heuristically in a di-allelic case, and by Sato in a multi-allelic case, neglecting any mutation factor. Our result extends Sato’s one.We next consider a continuum limit of the alleles space in the diffusion model. The limiting process is then a measure-valued diffusion process. Particularly if the alleles space is one-dimensional and the mutation operator is the Laplacian, we can derive an infinite dimensional stochastic differential equation of which solution defines a probabilitydensity-valued diffusion process.